Statistics for Analysis of Experimental Data
Catherine A. Peters
Department of Civil and Environmental Engineering
Princeton University
Princeton, NJ 08544
Published as a chapter in the
Environmental Engineering Processes Laboratory Manual
S. E. Powers, Ed.
AEESP, Champaign, IL
2001
1
Statistics for Analysis of Experimental Data
Catherine A. Peters
Department of Civil and Environmental Engineering
Princeton University
Princeton, NJ 08544
Statistics is a mathematical tool for quantitative analysis of data, and as such it serves as the
means by which we extract useful information from data. In this chapter we are concerned with data that
are generated via experimental measurement. Experimentation often generates multiple measurements of
the same thing, i.e. replicate measurements, and these measurements are subject to error. Statistical
analysis can be used to summarize those observations by estimating the average, which provides an
estimate of the true mean. Another important statistical calculation for summarizing the observations is
the estimate of the variance, which quantifies the uncertainty in the measured variable. Sometimes we
have made measurements of one quantity and we want to use those measurements to infer values of a
derived quantity. Statistical analysis can be used to propagate the measurement error through a
mathematical model to estimate the error in the derived quantity. Sometimes we have measured two
different things and we want to know whether there really is a difference between the two measured
values. Analysis of variance (t-tests) can be used to estimate the probability that the underlying
phenomena are truly different. Finally, we may have measured one variable under a variety of conditions
with regard to a second variable. Regression analysis can be used to come up with a mathematical
expression for the relationship between the two variables. These are but a few of the many applications of
statistics for analysis of experimental data. This chapter presents a brief overview of these applications in
the context of typical experimental measurements in the field of environmental engineering.
This chapter is necessarily brief in presentation. Students who seek a deeper understanding of
these principles should study a textbook on statistical analysis of experimental data. The bibliography at
the end of this chapter lists some useful textbooks, some of which are directly aimed at environmental
engineers and scientists.
Error Analysis and Error Propagation
Errors in Measured Quantities and Sample Statistics
A very important thing to keep in mind when learning how to design experiments and collect
experimental data is that our ability to observe the real world is not perfect. The observations we make are
never exactly representative of the process we think we are observing. Mathematically, this is
conceptualized as:
measured value = true value ± error
(1)
The error is a combined measure of the inherent variation in the phenomenon we are observing and the
numerous factors that interfere with the measurement. Every effort should be made to reduce systematic
errors through efforts such as calibration of measurement instruments. It is impossible to totally eliminate
all measurement error. If the underlying error is truly random (not biased) then we can still gain useful
information by making multiple observations (i.e. replicates) and calculating the average. In order for the
sample to be truly representative of the underlying phenomenon that is being measured it must be a
2
random sample. For example, let’s say that you are running an experiment in which you have set up
eight batch reactors and you plan to sacrifice one batch reactor every hour to measure the concentration of
some chemical. Every time you select a batch reactor you should randomly select from the remaining
reactors. You should not sample the reactors in the same order as you prepared them nor should you
sample the reactors in the order in which they are positioned on your bench top. You never know how
these other factors may influence the controlling processes in the reactors. By randomly sampling the
reactors, any systematic error due to other factors is randomly distributed across your measurements.
Randomness helps to ensure independence of the observations. When we say that we want “independent
observations” what we really mean is that we want the errors in the observations to be independent of
each other. Aside from nonrandom sampling, there are other laboratory activities that could jeopardize
independence of the observations. For example, if an inexperienced experimentalist gets better at making
a certain type of measurement, then the error may get smaller over time. In this case, the error is a
function of the order in which the measurement is made and the errors are not independent. Similarly, if a
measurement device wears out every time it is used then the error may increase over time. This too would
produce errors that are not independent. Random sampling and other efforts to make the observation
errors independent help to ensure representativeness. If all the observations are truly representative of
the same underlying phenomenon, then they all have the same mean and variance, i.e. the errors are
identically distributed. Sometimes the acronym IID is used to collectively refer to the criteria that a
sample of observations is independent (I) and identically distributed (ID).
Given a sample of n observations, the sample average is calculated as:
n
∑ xi
x = i =1
n
(2)
where xi represents the ith individual observation. The sample average is a statistic that is an estimate of
η, the mean, or central tendency, of the underlying random variable. The sample variance is
n
∑ (xi − x )2
s 2 = i =1
n −1
(3)
The sample variance is a statistic that is an estimate of the variance, σ2, in the underlying random
variable. Another useful statistic is the sample standard deviation, s, which is the square root of the
sample variance, σ. The quantity n-1 is the number of degrees of freedom associated with the sample
standard deviation.
It is often the case that we are more interested in the estimate of the mean than in the individual
observations. What we really want to know then is what is the variance in the average value. That is, how
does the variance in x translate into uncertainty in our ability to estimate the mean? The standard error
of the mean is
sx =
s
n
(4)
which also has n-1 degrees of freedom. Clearly, when the number of observations, n, is large, the
uncertainty in the estimate of the mean is small. This relationship demonstrates that there is more
uncertainty in an individual observation than in the estimated mean. Even if the underlying phenomenon
3
is quite variable and there are significant measurement errors, it is still possible to reduce uncertainty in
the estimate of the mean by making many measurements.
EXAMPLE. A student collects a series of twelve groundwater samples from a well. To
start, she measures the dissolved oxygen concentration in six of these. Her observations
in mg/L are:
8.8, 3.1, 4.2, 6.2, 7.6, 3.6
The sample average is 5.6 mg/L. The sample standard deviation is 2.3 mg/L. This value
can be interpreted as the error, or uncertainty, in any given measurement of the dissolved
oxygen concentration. Note that the variation in these data represent both natural
variation in the oxygen concentration in the water as well as variation due to
measurement error. The standard error of the mean is 0.95 mg/L. Notice that this is
considerably smaller than the sample standard deviation. After examining these statistics
the student decides that the uncertainty in the estimate of the mean is unacceptably large.
She proceeds to measure the dissolved oxygen concentration in each of the six remaining
samples. The additional observations in mg/L are:
5.2, 8.6, 6.3, 1.8, 6.8, 3.9
The grand average of all twelve observations is 5.5 mg/L and the standard deviation of
the sample of twelve observations is 2.2 mg/L. These statistics are comparable to those of
the smaller data set which provides some evidence that the original six observations are
representative of the underlying phenomenon. The new standard error of the mean is 0.65
mg/L. The reduction in the uncertainty in the estimate of the mean results from having a
larger number of observations in the sample.
The Normal Distribution
It is very often the case that an experimentalist will use a calculated sample average and standard
error to infer something about the probability of the random variable under observation or its relationship
to other random variables. To do this one must make an assumption about the shape of the probability
distribution of the errors in the experimental measurements. Most statistical techniques require an
assumption that the measurement errors have a normal probability distribution. The normal distribution
is also frequently called the Gaussian distribution. A plot of a probability distribution function (PDF)
for a normally distributed random variable x with mean of zero and standard deviation of unity is shown
in Figure 1a. For a given value of x, the value on the y axis is f(x), the probability density. The normal
PDF is symmetric, centered at the mean of x, and it extends from negative infinity to positive infinity. By
definition, the area under any probability distribution function equals unity. For a normal probability
distribution, 68% of the area under the curve lies within η ± σ, meaning that 68% of the total probability
is within one standard deviation of the mean. Practically speaking, one would expect that roughly 2/3 of
one’s observations would fall within this range. The area under the curve within η ± 2σ captures 95% of
the total probability and the area under the curve within η ± 3σ captures 99.7% of the total probability.
Another way to view the normal distribution is as a cumulative distribution function (CDF), shown in
Figure 1b. For a given value of x, the value on the y axis, F(x), is the cumulative probability associated
with values of the random variable less than or equal to x.
4
f(x)
(a) PDF
1
F(x)
(b) CDF
0.9
0.8
0.8
0.7
0.7
η=0
0.6
0.6
σ=1
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
-4
-3
-2
-1
1
0.9
0
x
1
2
3
4
-4
-3
-2
-1
0
x
1
2
3
4
Figure 1a and 1b. The PDF of the normal probability distribution (a) and the CDF of the
normal probability distribution (b) for a random variable x with mean of zero and
standard deviation of unity.
Note that the stipulation for application of many statistical techniques is that the errors in the
observations are normally distributed and not that the random variable itself is normally distributed. This
is an important distinction because many environmental variables have distributions other than the normal
distribution. For example, random variables that cannot assume negative values such as concentrations
and random variables that vary over orders of magnitude such as hydraulic conductivity of a porous
medium are typically lognormally distributed, i.e. the logarithm of the random variable is normally
distributed. Another positively skewed probability distribution function that is widely used to describe
environmental variables is the gamma distribution. For example, precipitation rates are often described
using the gamma distribution.
If numerous replications have been made for a given measurement, then it is possible to examine
whether the observations have a normally distributed error structure. This is typically done using a normal
probability plot. A normal probability plot is a form of the normal CDF in which the y axis has been
modified such that the cumulative distribution function appears to be linear. One can generate such a plot
by constructing a rank-ordered list of the observations, estimating the cumulate probabilities, and plotting
on special graph paper called normal probability paper (for more details see McBean and Rovers, 1998).
Despite the fact that a random variable may vary over time and space according to a non-normal
probability distribution, it is quite possible that the observation of a particular value of a non-normally
distributed random variable can be described using the normal distribution. As stated earlier, the error
represents both variation inherent in the random variable as well as measurement error. The latter arises
due to numerous small factors related to experimental design, sampling, detection, and analysis. For
example, the manufacturer of a thermometer may not have calibrated it very well so the temperature
markings do not exactly match the corresponding level of mercury. There may be impurities in the
mercury so that it does not expand and contract in a reproducible fashion. The experimentalist's line of
sight causes parallax error in reading the markings. The temperature varies slightly over the time period
of the measurement. The temperature varies spatially in the region where the experimentalist would like
to record the temperature. Collectively, these errors add up to generate the imprecision in an experimental
measurement. The central limit theorem says that as the number of variables in a sum increases the
distribution of the sum of random variables approaches the normal distribution regardless of the shape of
the distribution of the individual random variables. Experimental measurement error is the aggregate of a
large number of contributing errors. If the sources of error are numerous (as they usually are) then by the
central limit theorem we can say that experimental errors tend to have a normal distribution. Furthermore,
5
often we use statistical tools to make an inference about a sample average, which is a further summation
of values that are themselves likely to have normally distributed errors. This provides additional
justification to assume that sample averages have normally distributed errors.
It is often the case in experimental studies that we don't have enough observations to generate a
normal probability plot and make judgments about the shape of the underlying probability distribution. In
the example above, even with as many as twelve observations it may be difficult to judge the linearity of a
normal probability plot. Often we make only two or three replicate measurements which makes it
impossible to construct a meaningful probability plot. In these cases, we have to assume a normally
distributed error structure. Fortunately, the central limit theorem provides a theoretical basis for making
this assumption in experimental measurements.
Confidence Intervals
For any estimated statistic, such as a sample average, for which we have an estimated value and
an estimate of the standard error in that statistic, we can report confidence intervals. If the errors in the
measurement variable, x, have a normal probability distribution and if the observations are independent,
then the probability distribution for the error in the sample average, normalized by the standard error in
the sample average, is the t-distribution. The t-distribution is a symmetric probability distribution
centered at zero, like the normal probability distribution. The difference is that the t-distribution has a
variance that depends on the degrees of freedom of the standard error in the statistic of interest. Recall
that sx has n-1 degrees of freedom. If very few measurements have been taken, the number of degrees of
freedom is very small and the t-distribution has a very large variance.
The t-distribution is used to determine a t-statistic which is then used to calculate a confidence
interval for the true value of the mean, η. The t-statistic of interest is that which bounds a chosen level of
probability, 1-α, for the t-distribution with n-1 degrees of freedom. For example at a 90% probability
level 1-α = 0.90 and α = 0.10. Most statistics textbooks have tables of values of the t-statistic for various
levels of probability and values of degrees of freedom, and statistical software packages can compute tstatistics. The 1-α confidence interval for η is
x ± t n −1;a 2 s x
(5)
The reason the appropriate t-statistic is that which corresponds to the α/2 probability level is because that
value represents one side of a symmetric two-sided interval. We say that there is a 1-α probability that the
confidence interval contains the true value of η. Conventionally used probability levels are the 90%
(somewhat confident), 95% (fairly confident), and 99% (quite confident) probability levels.
EXAMPLE. For the oxygen concentration data discussed in the example above, what is
the 95% confidence interval for the mean? The standard error of the mean has 12-1=11
degrees of freedom. The t-statistic that corresponds to the t-distribution for 11 degrees of
freedom and 95% probability level is
t 11; 0.025 = 2.201
There is a 95% probability that the true value of the mean oxygen concentration lies
within the interval of
5.5 ± 2.201 ∗ 0.65 mg/L
5.5 ± 1.4 mg/L
4.1 to 6.9 mg/L
6
Notice that it is conventional to report standard errors, or values of the t-statistic
multiplied by a standard error, with no more than two significant figures. The magnitude
of the t n −1;a 2 s x term then dictates how many significant figures should be used to report
the value of the sample average. In this example, it would have been inappropriate to
report the sample average with three significant figures, i.e. as 5.51 mg/L, because that
implies a level of precision that is unwarranted given the uncertainty in the estimate. If
the value of the standard error had been 0.065 mg/L then the t n−1;α 2 s x term would
have been 0.14 mg/L. In this case, it would have been appropriate to report the sample
average with three significant figures.
Before we conclude this section, consider that sometimes a reported confidence interval does not
truly represent the uncertainty in the observation because the sample standard deviation may not capture
all the possible sources of variability in the observation. To precisely estimate uncertainty in a
measurement, an experimentalist must make replicate measurements. True replication involves
redundancy in all aspects of the experiment that may contribute to error. For example, suppose an
experiment is designed to infer a reaction rate by measuring the concentration of a reactant over time. The
experimentalist may set up a single reactor vessel and take samples for analysis at specified points in
time. To improve precision, the experimentalist may take more than one sample at a given point in time
and average the measured concentrations. There may be some variation due to small inconsistencies in
sampling and sample handling, but all these samples came from the same reactor. There are many more
possible sources of error that have not been captured by taking replicate samples from a single reactor.
The experiment itself has not been replicated. A better design would be to set up more than one reactor
vessel, and ideally these would be at different time periods and different spatial locations. Sometimes the
time and resources constrain the experimentalist’s endeavors in replication. If this is the case, the data
analyst must be cognizant of the extent to which the variation in a sample of observations represents the
true uncertainty in the measurement.
Estimation of Errors in Derived Quantities
Frequently, we make experimental measurements that are used to infer the value of a quantity that
is difficult to measure directly. For example, if we want to know the density of a fluid the easiest
approach may be to weigh a measured volume of the fluid. The density is calculated as the weight divided
by the volume. The question is how do the errors in the measurements of the weight and volume translate
into error in the estimate of the density?
Consider a random variable, z, that is a function of N random variables {x1, x2, …}. It can be
shown that if we assume the errors are relatively small and there is no covariance between the variables
{x1, x2, …} a Taylor series expansion of the error in z will produce the following expression for the
variance in z (for more details, see Bevington and Robinson, 1992)
σ z2
⎛ ∂z
= ∑ ⎜⎜
∂
i =1 ⎝ xi
N
2
⎞ 2
⎟⎟ σ x
i
⎠
(6)
This relationship can be used to estimate the variance in a derived quantity that is a function of
independent variables provided that the sample variances of the measured variables have been estimated.
Equation 6 also partitions the uncertainty in the independent variables which can provide useful
information about what measurements are the most important with regard to improving the precision in
the estimate of the derived quantity. Each independent variable, xi, contributes to the variance in z in two
ways, through its uncertainty i.e. σ xi , and through the mathematical sensitivity of z to xi, i.e. the partial
differential ∂z ∂xi .
7
Before proceeding, let’s examine the concept of covariance in experimental data. A covariance
of zero does not mean that x1 and x2 may not be interdependent, it simply means that the variation in one
is independent of the variation in the other. For example, we know that the density of a material can be
related to its weight and volume. The weight and the volume are most definitely related to each other. A
larger volume must weigh more and visa versa. However, there is probably no reason to believe that the
error in the measurement of the weight is in any way related to the error in the measurement of the
volume.
As an example of a derived quantity that is additively related to other variables, consider the case
that z is the weighted sum of two random variables, x1 and x2
z = ax1 + bx2
(7)
where a and b are constants. If the covariance between x1 and x2 is zero, then the variance in z is
σ z2 = a 2σ x21 + b 2σ x22
(8)
In general for a sum, z, that is the weighted sum of N independent variables
N
z = ∑ ai xi
(9)
i =1
where the ai’s are constant coefficients for the random variables, xi’s, the variance in z is
σ z2
N
= ∑ ai2σ x2i
(10)
i =1
EXAMPLE. An experiment is conducted to estimate the weight of water that has
evaporated from a small pan placed in the sunlight. The experimentalist weighs out 4.0
kg of water and places it in the pan, and repeats this process four more times. After a
period of time during which there has been no rain, the remaining water in the pan is
weighed and found to be 16.2 kg. The estimated weight of water that has been lost due to
evaporation, E, is related to amounts of water added and remaining through
E = 5A− R
where A is the weight of each individual amount of water that was added and R is the
weight of water that is remaining in the pan. The experimentalist calculates the estimated
value of E as 3.8 kg. Based on multiple measurements and past experience, the
experimentalist estimates that the device used to measure each individual amount of
water added has a standard deviation of 0.1 kg, i.e. s A = 0.1 kg . A different device was
used to measure the much larger weight of the remaining water. The measurement error
associated with this device is estimated to be s R = 0.2 kg . Assuming there is no
covariance in the measurements of A and R, the standard deviation of the estimate of E is
8
s E = 5 2 s A2 + (− 1)2 s R2
= 25 (0.01)+ 0.04
= 0.25 + 0.04
= 0.54 kg
It makes sense that the estimated error in the derived quantity, E, is greater than the error
in either of the variables that went into its calculation. From this calculation it is also
possible to examine which experimental measurement contributes the largest source of
uncertainty. In this case, the greatest uncertainty comes from the total measured weight of
the water added. Despite the fact that the measuring device used for this weight
measurement was more precise than that used for the measurement of R, the total
contribution to the uncertainty is larger due to the fact that this measurement had to be
made five times.
Quite often, derived quantities result from multiplicative relationships. Consider a random
variable, z, that is the product of two random variables, x1 and x2, each one raised to some power
z = x1a x2b
(11)
One can derive an expression for the variance in z by differentiating equation 11 to derive the partial
differentials of z with respect to x1 and x2, and then applying equation 6. Alternatively we can turn
equation 11 into an additive relationship by taking the logarithm of both sides.
ln z = a ln x1 + b ln x2
(12)
Assuming x1 and x2 are independent (i.e. no covariance), we can apply the principles in equation 10 to the
summation and get
σ ln2 z = a 2σ ln2 x1 + b 2σ ln2 x2
(13)
The standard deviation of the natural logarithm of a random variable is approximately equal to the
relative standard error (also called the coefficient of variation), i.e.
σ ln x ≈
σx
x
(14)
if the standard deviation for that variable is small. This approximation should look familiar to those who
have studied calculus since the total differential for the logarithm of a variable is
d ln x =
dx
x
The approximation in equation 14 holds for relative standard errors less than 10%, i.e. σ x x ≤ 0.10 , and
is a good approximation even for relative standard errors up to 20%. Applying the approximation in
equation 14 to equation 13 gives
9
σ z2
z2
=a
2
σ x2
x2
+b
2
σ 2y
y2
(15)
For multiplicative relationships, the error in a derived quantity depends not only on the errors in the
independent variables and the sensitivities but also on the magnitudes of the independent variables. In
general for a variable, z, that is the product of powers of N independent variables:
N
z = ∏ xi ai
(16)
i =1
the relative variance in z is
σ z2
z2
N
=∑
i =1
ai2
σ x2i
(17)
xi2
In the event that replicate measurements of x have been made, the coefficient of variation, σ X x , is
estimated by the relative standard error which is s X x .
EXAMPLE. An experiment is designed to infer the density of a liquid (ρ) from its
measured mass (M) and volume (V). How do the errors in M and V translate into error in
the calculated value of ρ ?
ρ=
M
V
ln ρ = ln M − ln V
σ ln2 ρ = σ ln2 M + σ ln2 V
If the errors in M and V are not excessively large,
σ ρ2 σ M2 σ V2
=
+
ρ2 M 2 V 2
2
2
σρ
⎛σ M ⎞ ⎛σV ⎞
= ⎜
⎟
⎟ +⎜
ρ
⎝ M ⎠ ⎝V ⎠
The relative standard error s M M is an estimate for σ M M , and likewise for V.
Suppose that the relative standard error in M is 0.01 (i.e. 1%), and the relative standard
error in V is 0.05 (i.e. 5%). What is the relative standard error in ρ?
sρ
ρ
=
(0.01)2 + (0.05)2
= 0.05
The error in the inferred value of the density is 5%. In this case, the error in the density
estimate is largely controlled by uncertainty in the measured volume. Efforts to reduce
the error in the volume measurement would be the most fruitful way of improving the
precision in the estimated density.
10
Hypothesis Testing and the t-Test
One of the most common uses of statistics is to compare a measured value with either a known
value or another measured value. For example, suppose an experiment is being conducted to examine the
extent to which solution chemistry affects the dissolution rate of a particular mineral. Let’s assume there
are two conditions of interest. There are two general ways of designing such an experiment. One
possibility is to set up replicates of experimental systems under one condition and an independent set of
replicate experimental systems that are observed under the other condition. The average of the measured
dissolution rates from each set would be assumed to represent the dissolution rate under the respective
condition. In this case, the experimentalist would be interested in comparing the two averages and
inferring the extent to which the two values differ. Alternatively, a series of experimental systems could
be set up all with one condition and the dissolution rates measured. Then the solution chemistry could be
changed in each system and the dissolution rate measured again. The two measurements for each
experimental system are clearly not independent of each other and cannot be analyzed separately. In this
case, the experimentalist would examine the change in dissolution rate for each experimental system, and
compute the average of the differences in the dissolution rate. This average would be compared with zero
(a known value). Obviously, there are technical constraints that would favor one experimental design over
the other, but both of these experimental designs would allow one to examine the effect of solution
chemistry on mineral dissolution rate. It is important for the experimentalist to understand the difference
in the appropriate statistical procedure for data analysis in each case.
Statistical analysis that examines differences between samples of observations is called analysis
of variance (ANOVA). Analysis of variance usually refers to statistical analysis involving simultaneous
comparison of multiple sets of observations, not just the comparison of two averages. When applied to
comparison of two averages, or an average and a known value, the statistical procedure known as
ANOVA simplifies to what is commonly called a t-test. For detailed discussions of ANOVA and
experimental design involving multiple sets of observations, the reader is referred to Box, et al. (1978).
Here we will discuss the simple case of t-tests for comparison of two values. In statistical t-tests, and in
other statistical tests, the first step is to formulate the null hypothesis. For the case of comparison of two
values, it is conventional that the null hypothesis be a statement that there is no difference between the
two values. Then we analyze the data to examine the extent of the evidence to reject the null hypothesis
relative to a specific alternate hypothesis.
Comparing A Sample Average with a Known Value
The example given above in which mineral dissolution rates are observed in replicate
experimental systems in which the solution chemistry is changed is a case where a sample average is
compared with a known value. This kind of experimental design is often called a “paired comparison”.
The average of the differences in the pairs of observations is compared with the value zero. Another case
where one would need to compare a sample average with a known value would be, for example, if the
concentration of a chemical in a solution is known with great certainty, and we are testing whether an
instrument generates measurements that are consistent with the known concentration. In either case, the
null hypothesis is that the true sample mean, η, is the same as the known value, η0. Of course, the
estimated mean, i.e. the sample average x , will most likely be different from η0. If there is a great deal of
uncertainty in the measured sample average then there may be insufficient evidence to reject the null
hypothesis.
We assume that the errors in the measured variable are IID and normally distributed. Because x
has uncertainty, one can think of it as a random variable. The error in this random variable, i.e. the
difference between the sample average and the true sample mean, has the t-distribution with n-1 degrees
of freedom, scaled by the standard error of the sample average, i.e.
11
(x − η ) ~ t n−1 s x
(18)
where n is the number of observations used to compute the average. If the sample used to compute x is a
set of paired differences, then n is the number of sets of pairs, i.e. the number of differences that are used
to compute the average difference. The symbol “~” in equation 18 means “is distributed as”. As is
consistent with the null hypothesis, the difference in equation 18 has an expected value of zero.
To test the null hypothesis we estimate the probability associated with an alternate hypothesis.
Possible alternative hypotheses are η ≠ η0, η < η0, or η > η0. The choice of the alternate hypothesis
determines whether we conduct a one-sided t-test or a two-sided t-test. Imagine that we are interested in
the alternate hypothesis that the true mean is greater than the known value, i.e. the difference (η − η 0 ) is
greater than zero. This means we want to conduct a one-sided t-test. The probability associated with this
alternate hypothesis is:
Pr{η − η 0 > 0}= Pr{η > η 0 }
= Pr{(x − η ) ≤ (x − η 0 )}
⎧⎪⎛ x − η ⎞ ⎛ x − η 0
⎟≤⎜
= Pr ⎨⎜⎜
⎪⎩⎝ s x ⎟⎠ ⎜⎝ s x
⎧⎪
⎛ x −η0
= Pr ⎨t n −1 ≤ ⎜⎜
⎪⎩
⎝ sx
= 1−α
⎞⎫⎪
⎟⎟⎬
⎠⎪⎭
⎞ ⎫⎪
⎟⎟ ⎬
⎠ ⎪⎭
(19)
The term in parentheses is called the “observed” value of the t-statistic. It is a ratio of the observed error
in the sample average (assuming the null hypothesis to be true) to the standard error, and it can be thought
of as a ratio of the “signal” to “noise”. If the difference between x and η0 is large relative to the standard
error in x , then the probability 1 − α is large. In this case, it is very unlikely that the observed difference
would occur due to random chance. One would say that there is significant evidence to reject the null
hypothesis. Alternatively, we could have estimated the probability of the alternate hypothesis that the true
mean is less than the known value. Both of these cases are one-sided t-tests. For the third alternate
hypothesis, that the true mean is different from (either greater than or less than) the known value, we
conduct a two-sided t-test. Because of the symmetry of the t-distribution, the probability associated with
this is:
⎧⎪ x − η 0
x − η0
Pr ⎨−
≤ t n −1 ≤
sx
sx
⎪⎩
⎫⎪
⎬ = 1−α
⎪⎭
(20)
If the probability 1 − α is large then there is a great deal of evidence that the true mean is different from
η0. If the probability 1 − α is small then there is very little evidence to reject the null hypothesis and we
say that there is not a statistically significant difference between η and η0. The two-sided t-test is actually
the most common form of the t-test because often one does not have a priori knowledge of the sign of the
difference between η and η0.
Keep in mind that there is a distinction between a difference that is statistically significant and a
difference that is important. For example, one may find that there is a statistically significant difference
between mineral dissolution rates at different solution chemistry conditions, but perhaps the magnitude of
12
this difference is too small to be important relative to other rate processes. The t-test can only indicate
whether differences are statistically significant.
EXAMPLE. It is believed that increased partial pressure of carbon dioxide, PCO2, in the
atmosphere may accelerate weathering of minerals by increasing the concentration of
carbonate in waters and by affecting the pH. An experiment is conducted to examine how
variation in PCO2 affects the rate of dissolution of calcium from soil minerals. The
experiment is designed by setting up a series of 3 replicate reactor vessels. The systems
are identical with regard to all the factors that might affect the mineral dissolution
process. Initially, the PCO2 is controlled at 600 Pa and the rate of Ca dissolution is
determined. Then the PCO2 is adjusted to 1000 Pa in each of the reactor vessels and the
rate of Ca dissolution is again determined. To compare the dissolution rates for each
vessel, one must assume that the treatment at 600 Pa did not change the system in a
significant way such that the initial conditions for the 1000 Pa treatment can be assumed
to be the same as the initial conditions for the 600 Pa treatment. (Sometimes it is not
possible to design experiments like this.)
The surface area-normalized dissolution rate measurements are shown in the following
table. The differences between the rates for each vessel are tabulated in the last column,
and the averages, sample standard deviation of the differences, and standard error of the
mean differences are also shown.
Ca Dissolution Rate [μmol m-2 hr-1]
Reactor vessel
1
2
3
Averages
A; PCO2 = 600 Pa
347
96
174
205.7
B; PCO2 = 1000 Pa
600
337
402
446.3
Difference
B-A
253
241
228
240.7
Sample standard deviation
of the differences
12.5
Standard error of the
mean difference
7.2
What is the probability that there is a difference between the true dissolution rates at the
two PCO2 conditions? This question calls for a two-sided t-test. This is appropriate if we
can assume that the observed differences are IID with normally distributed errors.
To compute the probability in equation 20, we must compute the probability that the tstatistic with (3-1) = 2 degrees of freedom lies within the range bracketed by the positive
and negative values of the observed t-statistic:
x − η0
240.7 − 0
=
= 33
7.2
sx
In this case, the “known value” against which the average is being compared is zero. The
observed value of the t-statistic, 33, is very large indicating a strong signal to noise ratio.
One can use a table of t-statistics to find the probability that t2 lies within the range of –33
to +33, but most textbooks list values of the t-statistic for only a few selected values of α
(e.g. 0.25, 0.1. 0.05, 0.025, 0.01, 0.005). It is more useful to compute the actual
13
probability associated with the “observed” value. In Microsoft® Excel, for example, one
can compute this probability using the TDIST function. For this experiment,
Pr{− 33 ≤ t 2 ≤ 33} = 0.999
In words, the probability that the true dissolution rates at the two PCO2 conditions are
different is 99.9%. There is only a 0.1% probability that this difference would occur by
random chance. These findings present compelling evidence that the mineral dissolution
rate is dependent on PCO2.
Comparing Two Sets of Measurements
Consider an experimental design in which mineral dissolution rates are observed in two series of
replicate experimental systems where the solution chemistry for all the systems in one series is fixed to
represent one condition and for all the systems in the other series is fixed to represent the other condition.
This is a case where the dissolution rate measurements under the two conditions are independent of each
other and the two sample averages are compared with each other. Often times in this type of experimental
design, one of the experimental conditions is viewed as the “control”. This type of experimental design is
used even in the case where we think we know a priori how the control system will behave. For example,
suppose the mineral dissolution rate of interest has been reported in the literature for a baseline condition,
and the objective of our experiment is to examine the effect of changing the solution chemistry.
Conceivably we could run replicate experiments only at the new solution chemistry and compare the
average mineral dissolute rate with the literature value. This is usually a bad idea. A good experimentalist
always runs parallel experiments, i.e. control systems, that are identical to the other experimental systems
in all ways except for the treatments of interest. This eliminates the additional variation that could arise
due to conducting the experiments at different times and in different labs.
Let the subscripts “A” and “B” denote the two experimental conditions, and ηA and ηB denote the
true mean values of the observation variable, x, for the two conditions, respectively. The null hypothesis
is that the difference between ηA and ηB is zero. The purpose of a t-test in this type of experimental
design is to examine whether the difference between the observed sample averages, x A and x B , is large
relative to the uncertainty in the averages.
B
B
We assume that the errors in the measurements for the two experimental conditions are IID and
normally distributed. Just as one can consider x to be a random variable, one can consider the difference
x A − x B to be a random variable. The error in this difference has a scaled t-distribution:
(x A − x B ) − (η A − η B ) ~ tν
s ( xA − xB )
(21)
where the scaling factor for the t-distribution is the standard error in the difference of the two sample
averages. Because the null hypothesis is that the difference between ηA and ηB is zero, equation 21 can
also be written
B
(x A − x B ) ~ tν s ( x − x
A
B
(22)
)
From equation 10 we see that the variance of a difference between two variances is the sum of the
variances of the variables:
σ (2x
A − xB )
= σ x2 + σ x2
A
B
(23)
So the standard error in the difference is:
14
s A2 s B2
s ( xA − xB ) =
+
nA nB
2
(24)
2
where s A and s B are the sample variances for the two experimental conditions. If the magnitude of the
errors in the measurements of xA and xB are approximately equal, then the degrees of freedom associated
with this standard error (ν in equation 22) is the sum of the degrees of freedom for the standard errors in
each of x A and x B , i.e. ν = nA + nB – 2, where nA and nB are the numbers of values in the samples of
observations from experimental conditions A and B, respectively. The common variance (also called the
pooled variance) is estimated as a weighted average of the variance in xA and xB:
B
B
B
B
s2 =
(n A − 1)s A2 + (n B − 1)s B2
(25)
nA + nB − 2
The standard error in the difference of the averages can then be written:
s ( xA − xB ) = s
1
1
+
nA nB
(26)
where s is the common standard deviation computed as the square root of s2.
To test the null hypothesis that the true means are the same, we estimate the probability
associated with one of the following possible alternate hypotheses: ηΑ ≠ ηB, ηΑ < ηB, or ηΑ > ηB. A twosided t-test is used to test the alternate hypothesis that the true means are different from (either greater
than or less than) each other. The probability associated with this is:
B
⎧⎪ x − x
x − xB
B
Pr ⎨− A
≤ tν ≤ A
s ( x A − xB )
⎪⎩ s( x A − xB )
B
⎫⎪
⎬ = 1−α
⎪⎭
B
(27)
If the observed difference is large, the probability 1 − α will be large. However, in order for this
difference to be statistically significant, it must be large relative to the standard error of the difference. If
there is very little precision in the estimates of the averages then one still may not be able to say that that
difference is statistically significant even if the observed difference is large.
EXAMPLE. An experiment is conducted to examine how PCO2 affects the rate of
dissolution of calcium from soil minerals. As in the previous example, there are two PCO2
conditions of interest, but in this case the experimental design involves two independent
series of experimental systems. Three reactor vessels are operated at 600 Pa and three
separate reactor vessels are operated at 1000 Pa. Imagine, for demonstration purposes,
that the mineral dissolution rates that are observed in these six systems are the same
values as those in the previous example.
15
Ca Dissolution Rate [μmol m-2 hr-1]
A; PCO2 = 600 Pa
347
96
174
205.7
B; PCO2 = 1000 Pa
600
337
402
446.3
Sample standard deviation
128.5
137.0
Sample variance
16502
18766
Average
The magnitude of the observed difference in the averages is:
x A − x B = 240.7 μmol m -2 hr -1
What is the probability that there is a difference between the true mean dissolution rates
at the two PCO2 conditions? As with the previous example, a two-sided t-test is
appropriate. This is appropriate with an assumption that the errors in the observed
mineral dissolution rates are IID and normally distributed.
The observed t-statistic in equation 27 requires the standard error of the difference in the
averages. Because we have assumed the underlying measurement errors for conditions A
and B are “identically distributed”, this implies that the variances are equal. The
estimated common variance and common standard deviation are:
s2 =
(3 − 1) *16502 + (3 − 1) *18766 = 17634
3+3−2
s = 17634 = 132.8 μmol m -2 hr -1
The standard error in the difference of the averages is:
s( xA −xB ) = 132.8
1 1
+ = 108.4 μmol m -2 hr -1
3 3
The degrees of freedom associated with this standard error is ν = 3 + 3 − 2 = 4 . With a
null hypothesis that the true difference between the means is zero, the observed t-statistic
is:
x A − xB
240.7
=
= 2.22
s ( x A − xB )
108.4
The probability of the alternate hypothesis is estimated as:
Pr{− 2.22 ≤ t 4 ≤ 2.22} = 0.91
In words, the probability that the true dissolution rates at the two PCO2 conditions are
different is 91%. There is a 9% probability that this difference would occur by random
chance.
16
Comparing the Two Types of Experimental Designs
Before concluding this section, let’s compare the two types of experimental designs described
above. Two sources of variation determine the observation of mineral dissolution rates. First there is the
variation that is caused by the change in solution chemistry. Second there is the variation caused by
random error that produces different observations for replicate experiments. A good experimental design
is one that maximizes the variation due to change in solution chemistry (the “signal”) relative to the
variation due to random error (the “noise”). If the experiment is conducted in a way in which each
replicate experimental system is observed first at one condition and then at the other condition then the
variation across replicate experimental systems is blocked from the observed variable. Imagine that one
of the reactor vessels is closer to the window where the temperature is slightly colder than the average
room temperature. The mineral dissolution rates that are measured for that vessel may be biased due to
this temperature difference relative to the other vessels, but this will be true for the mineral dissolution
rates at both conditions. The difference in mineral dissolution rates is likely much less sensitive to
temperature than the actual rates. This is demonstrated in the example of the paired comparisons. The
variation in the observations for a given PCO2 condition is fairly large. Notice that the ranges of the
observations for the two PCO2 conditions overlap. However, there is much less variation in the differences
in the dissolution rates. This experimental design reduces the effect of variation across replicate
experimental systems on the inferred difference between the two treatments. In the paired comparison, the
difference between the means is much more statistically significant than in the case of the comparison
between two averages from independent series of experiments. In general, given a choice between these
two types of experimental designs, paired comparisons should be used if other technical constraints don’t
preclude this design. For additional discussion on the advantages of paired comparisons, see Berthouex
and Brown (1994).
Linear Regression Analysis
Statistical Regression in Mathematical Modeling
We often wish to use experimental data to develop a mathematical model between two or more
variables. We can distinguish between two applications of statistics for mathematical modeling. First, if
we have a variable y that we believe to be related to a variable x then we could make a series of n
measurements of the pair {xi, yi} where i is an index that runs from 1 to n. If we have no a priori
knowledge of the possible mathematical relationship between y and x, then we can use statistical
techniques to estimate a mathematical relationship that captures the variation of y with x. Such a
mathematical model is called a statistical model, which is purely empirical. An example of a useful
empirical model is a power-series polynomial.
y = a1 + a 2 x + a3 x 2 + L + a m x m−1
(28)
where {a1 … am} are the parameters of the function.
It is more common that a mathematical relationship between the variables is known (or
postulated) based on theoretical principles. For example, based on past experience and the principles of
chemical thermodynamics, we know that in a system containing water in equilibrium with an organic
liquid phase (e.g. octanol), for dilute concentrations the concentration of a chemical in the water phase, y,
is linearly related to the concentration of that chemical in the organic liquid phase, x
y = ax
(29)
17
In this case, a is the model parameter describing the proportionality between the two concentrations.
Another example of a theoretical model is the Freundlich isotherm which is often used to describe
chemical equilibrium in a system of water in contact with a sorbing medium such as soil or sediment. The
Freundlich model relates the concentration of a chemical in the water phase, y, to the concentration of that
chemical in the sorbing medium, x, according to the following function
y = a1 x a2
(30)
This model has two parameters, a1 and a2.
Both of these applications of statistics are examples of regression, in which a mathematical
model has been established (either with a theoretical basis or not) and statistical regression is used to
estimate the parameters of the model based on a set of data. The examples given above are for the case
where there is one dependent variable y and one independent variable x. The mathematical modeling
problem may involve multiple dependent variables and/or multiple independent variables. The case of
multiple dependent variables can be a difficult problem and involve complex regression techniques. The
case of multiple independent variables is a simpler problem and often can be handled with little more
complexity than the case of a single independent variable. In general, the mathematical model we
consider here is
y = f ( x1 , x 2 , K , xl ; a1 , a 2 , K , a m )
(31)
where y is the single dependent variable, x1 through xl are the l independent variables, and a1 through am
are the m model parameters. The general regression problem is this: given a mathematical model and a set
of data, i.e. a series of n observations of the set {x1i , x 2i , K , xli , yi } , what are the values of the
parameters {a1 , a 2 , K , a m } that produces a model that best describes the relationship between the
dependent and independent variables. Statistical regression is also referred to as model calibration, data
fitting, data reduction, or parameter estimation. Sometimes the phrase “curve fitting” is used specifically
in reference to statistical regression of data to a polynomial function.
Linear Least Squares Regression
In the previous section we defined statistical regression as an activity that produces a model that
“best describes the relationship” between y and the independent variables. Conventionally, the criterion
we use to define “best” derives from the method of maximum likelihood. In words, the method of
maximum likelihood seeks the values of the parameters that maximize the probability of having obtained
the set of observations in hand. If we assume that the errors in y are independent and normally distributed,
the method of maximum likelihood leads to a goodness-of-fit parameter, Χ2 (“chi-squared”), which is
defined as
⎛ y − f (x1i , x 2i ,K, xli ; a1 , a 2 ,K, a m ) ⎞
⎟⎟
Χ ≡ ∑ ⎜⎜ i
σi
⎠
i =1 ⎝
2
n
2
(32)
where σi is the standard deviation associated with the ith value of y. The “best” values of the parameters
are those that minimize Χ2. It is quite often the case that we do not know σi. If we make a further
assumption that the observations of y are identically distributed, i.e. not only do all values have the same
underlying error structure (in this case a normal distribution) but they also have the same magnitude of
error, then the value of σi is the same for all i. Taken together, the three assumptions are stated concisely
as the errors are IID and normally distributed. With this assumption, Χ2 becomes
18
Χ2 =
1
σ
2
n
∑ ( yi − f (x1i , x2i ,K, xli ; a1 , a2 ,K, am ))2
(33)
i =1
where σ is the common standard deviation in y. Minimizing Χ2 is now a matter of finding the values of
{a1 , a 2 ,K, a m } that minimize the summation on the right hand side of equation 33. This form of
statistical regression is called least squares regression.
Experimentalists should be aware of the times when it is not appropriate to assume that all the
measurements have the same standard deviation. If one is using an instrument for which the error is
proportional to the magnitude of the measurement then the errors will be small for measurements of
smaller values and vice versa. Similarly, if the data to be used in the regression were obtained via
different measurement techniques then the errors can be substantially different. For example, if low
values of aqueous concentration are measured using a calibration curve designed specifically for a low
concentration range, and high values of aqueous concentration are measured using a calibration curve
designed for a high concentration range, then it is quite likely that they two sets of observations will have
different measurement errors. The errors may be constant in a relative sense (e.g. the % error), but the
absolute errors may be very different. After the regression has been performed, one can test the
assumption of common errors by examining a plot of the residuals (i.e. estimated errors) versus the x
variable. The residuals should have a random appearance and not appear to be related to the value of x. If
the experimentalist has good reason to be concerned about the lack of a common error in the observations
of y then the extra effort should be taken to estimate the errors in the observations. In this case, regression
should be performed by minimizing the Χ2 function in equation 32. This is a form of weighted least
squares regression.
If the function f ( x1i , x 2i , K , xli ; a1 , a 2 , K , a m ) is linear in the parameters then the summation
in equation 33 can be analytically differentiated with respect to each of the parameters. Setting these
differentials equal to zero produces m linear equations which can be solved simultaneously for the values
of the parameters that minimize Χ2. This form of statistical regression is called linear least squares
regression. Note that the model function need not be linear in the independent variables in order to use
linear least squares regression. The polynomial function shown in equation 28 is nonlinear in x but it is
linear in the parameters. Consequently this function can be easily differentiated to come up with
analytical expressions for the optimum values of the parameters. An example of a function that is
nonlinear in the parameters is the Freundlich isotherm (equation 30). In its current form, this model
cannot be fit using linear least squares regression. A transformation of the Freundlich equation generates a
form that is linear in the parameters, although nonlinear in the variables:
log y = log a1 + a 2 log x
(34)
The parameters of this equation, i.e. log a1 and a2, can be estimated using linear least squares regression
by regression to the transformed data {log xi, log yi}. However, caution is advised when linearizing
models for the purpose of conducting linear least squares regression. The assumption that must be
satisfied in this case is that the errors in the transformed variables are IID and normally distributed. If the
errors in the untransformed variables {xi, yi} are IID and normally distributed then the errors in the
transformed variables most certainly are not. If this is the case, then such models should be calibrated
using nonlinear regression techniques.
Consider the simplest case where y is linearly related to a single independent variable x,
y = a1 + a2 x
(35)
19
Because this function is linear in a1 and a2, linear least squares regression can be applied. (The function
also happens to be linear in x, but this is not a necessary condition for linear least squares regression.) The
Χ2 is
Χ2 =
n
( y − a1 − a2 xi )2
2 ∑ i
σ
1
(36)
i =1
Differentiating equation 36 with respect to a1 and a2 and setting these differentials equal to zero produces
the following equations for the optimum values of the parameters:
a1 = y − a2 x
a2 =
∑ xi y i − n x y
∑ xi2 − n x 2
(37)
(38)
where the summations are taken from i=1 to n. Most experimentalists need not use equations 37 and 38
directly as there are numerous mathematical and statistical software packages that have these functions
built-in. For example, in Microsoft® Excel these functions can be applied to a sample of data using the
“Regression” module in the “Data Analysis” tool which can be used for single-variable or multi-variable
linear least squares regression.
One of the assumptions that is implied in the application of the maximum likelihood approach to
statistical regression is that there is error only in the y variable and there is no error in the x variable(s). In
experimentation it is quite often the case that both x and y are measured variables and thus both are
subject to error. The x variable may be something that has negligible error, like a measurement of time or
length that can be recorded precisely. If there is appreciable error in the estimate of the x values, part of
the uncertainty in the y value is due to the fluctuation of x and its effect on the value of y. Thus the error
in the estimate of y at a given x value is the sum of the measurement error for y plus the error propagated
by the uncertainty in x. We can still apply least squares regression if the portion of the uncertainty in y is
small relative to its own measurement error. This is equivalent to saying that the x value is known to
much greater precision than the y value. One should always perform statistical regression by assigning the
variable with the greatest measurement error as the dependent variable, or the “regression variable”, even
if it means writing the mathematical function in a form that is inverse to its conventional presentation.
EXAMPLE. An experiment is conducted to determine the Freundlich isotherm
parameters for sorption of tetrachloroethylene onto a particular soil sample. Written using
conventional notation, the Freundlich equation is
q = K FC n
where the variables are q, the mass of solute sorbed per unit mass of soil, and C, the
solute concentration in the aqueous phase. The parameters are KF, the capacity coefficient
and n, the exponent constant. Assume that log-transformed values of the variables have
errors that are IID and normally distributed. This allows linear least squares regression
using the linearized model:
log q = log K F + n log C
A series of ten experimental systems are set up which are identical in all respects except
for the mass of tetrachloroethylene added. After equilibration under constant temperature
conditions, the solute concentration in the aqueous phase is measured and the mass of
solute associated with the solid phase is determined. The measurement error in q is
20
believed to be much larger than the measurement error in C so it is appropriate that ln q is
the dependent variable, i.e. the “regression variable”. The data are shown in the table
below.
i
1
2
3
4
5
6
7
8
9
10
C [μg/L] q [μg/g]
9.99
21.6
44.9
102.3
216.1
463.5
986.0
2130
4538
9852
0.512
0.435
2.29
2.37
5.04
11.6
19.9
22.4
140.0
122.9
log C
log q
1.00
1.34
1.65
2.01
2.33
2.67
2.99
3.33
3.66
3.99
-0.291
-0.362
0.359
0.375
0.702
1.065
1.298
1.350
2.146
2.090
The “Regression” tool in Microsoft® Excel was used to estimate the model parameters.
A portion of the output is shown below.
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.9809162
R Square
0.962196592
Adjusted R Square
0.957471166
Standard Error
0.181425915
Observations
10
ANOVA
df
Regression
Residual
Total
Intercept
X Variable 1
1
8
9
SS
MS
F
6.702263327 6.702263327 203.6211
0.2633229 0.032915363
6.965586228
Coefficients Standard Error
t Stat
P-value
-1.26712979
0.160599963 -7.8899756 4.82E-05
0.857214638
0.060072839 14.26958761 5.67E-07
From the column labeled “Coefficients”, we find the estimated parameter values:
KF = 0.054 L/g
log KF = -1.27
n = 0.86
The following plot shows the measurements of q and C plotted in logarithmic space along
with the regression line.
21
1000
q [microg/g]
100
10
1
0.1
1
10
100
1000
10000
C [m icrog/L]
The plot above shows no error bars on the data points. That is because the error in an individual
measurement of log q is unknown. In order to estimate the error for a given value of q, one would need to
set up replicate experimental systems with the same total mass of tetrachloroethylene. This kind of
replication is rarely done when data are collected for regression purposes. It is possible that to determine
the measured value of q, the experimentalist might take multiple samples from each experimental system
or analyze a given sample more than once. These actions improve precision of the measurement but the
variance in these measurement do not represent the true variance. Sometimes error bars that are shown in
plots of experimental data are based on sample replications or only the analytical instrument. It is OK to
put such error bars on a plot of data provided that it is explicitly stated that these error bars likely under
represent the true error. The true error can be determined only by replication of the experiment.
Regression analysis can be used to estimate the error in an individual measurement of the
dependent variable. The variance in the y variable that has not been explained by the regression model is
the residual sum of squares, SR:
n
S R = ∑ ( yi − f ( x1i ,K, xli ; a1 ,K, am ))
2
(39)
i =1
From equation 33, we see that this is the quantity that is minimized in the regression so the parameters in
equation 39 are the best-fit values. For a two-parameter linear model as in equation 35,
n
S R = ∑ ( y i − a1 − a 2 xi )
2
(40)
i =1
In Microsoft® Excel regression output, this quantity is found in “SS” column in the row labeled
“Residual”. The degrees of freedom associated with this summation is the number of observations minus
the number of parameters, n-m. In Microsoft® Excel regression output, this quantity is found in the “df”
column in the row labeled “Residual”. Dividing the residual sum of squares by the residual degrees of
freedom gives a quantity that has an expected value of σ2, the common variance of measurements of the
dependent variable. This is predicated on an assumption that the parameterized model is an accurate
representation of the true underlying relationship between y and x. This quantity, s2, is called the residual
mean square (also called “mean squared error”):
22
s2 =
SR
n−m
(41)
In Microsoft® Excel regression output, this quantity is in the “MS” column in the row labeled “residual”.
The square root of s2 gives an estimate of the common standard deviation of measurements of y.
EXAMPLE. For the regression example above, the residual sum of squares and residual
mean square are obtained from the regression output
S R = 0.263
s2 =
0.263
= 0.033
8
Assuming the calibrated model is an accurate representation of the relationship between
log q and log C, the square root of the residual mean square is an estimate of the standard
deviation in measurements of log q.
s = 0.033 = 0.18
Keep in mind that the assumption made about the measurements in log C is that they are
error-free.
If the value of s2 is known a priori or it is estimated from replicate experiments for fixed values
of the x variable, then the residual mean square can be compared to this value to examine the extent to
which the model fits the data. The statistical test to compare these two variances is an F-test, and in this
context it is called a lack-of-fit test, or a “goodness-of-fit test”. Without an independent estimate of s2
one can say nothing about whether the data fit the model, although this is a commonly made
misinterpretation of the regression statistics.
An alternative, and very popular, means of quantifying the extent to which the variation in the
data have been captured by the model is to compute the R-square and the adjusted R-square. The R-square
is the ratio of the variance in y measurements that is explained by the fitted model, i.e. the regression sum
of squares (SM), to the overall variance of the y measurements, i.e. the total corrected sum of squares (SD).
Both quantities are found in the Microsoft® Excel regression output in the “SS” column. Because the
total corrected sum of squares is the sum of the SM and SR, the R-square can be written in terms of SR and
SD.
R2 =
SM
S
= 1− R
SD
SD
(42)
When R2 is close to unity then we say that a large portion of the overall variation in the data are explained
by the model. It means that y is related to x, or highly correlated to x. It does not mean that the model that
has been specified is the correct model. The only way to examine if a model is the correct representation
of the underlying functional relationship is with an independent estimate of σ2 and a lack-of-fit test.
The adjusted R-square takes into account the degrees of freedom associated with each sum of
squares:
2
RADJ
= 1−
S R / (n − m )
s2
= 1−
S D / (n − 1)
S D / (n − 1)
(43)
This term is often reported when doing statistical regression using polynomial models. When the modeler
has a choice of the number of parameters in the model, then it is possible to improve the regression
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merely by increasing the number of parameters. Often this activity results in meaningless models that
“overfit” the data. The idea behind the adjusted R-square is that it has a penalty term for the number of
parameters. If m is large then the residual degrees of freedom n-m may be small. The adjusted R-square
will reflect the fact that the a large portion of the overall variance has been explained only because of the
large number of parameters. The R-square will not reflect this.
EXAMPLE. For the regression example above, the R-square can be computed from the
values in the “SS” column, or it can be read directly from the “Regression Statistics”:
R2 =
6.70
0.26
= 1−
= 0.96
6.97
6.97
Typically R-square values greater than 0.90 are considered indicators of a high degree of
correlation between the variables. This R-square of 0.96 indicates that log q and log C are
highly correlated. Note that the adjusted R-square (also in the “Regression Statistics”) is
very close to the R-square. For this model the number of parameters is small, and the
residual degrees of freedom, 8, is almost equal to the total degrees of freedom, 9.
Estimation of the Errors in the Parameters
Because the estimated parameters in linear least squares regression are linear functions of
normally distributed variables (equations 37 and 38), both a1 and a2 in equation 35 can be considered
normally distributed random variables. As such, for each parameter, the error in the parameter divided by
its standard error is a random variable that is distributed according to the t-distribution. The formulas for
computing the standard errors in the fitted parameters are given in most textbooks. These quantities can
be found in the Microsoft® Excel regression output in the “Standard Error” column next to the estimated
parameter values. The degrees of freedom associated with both standard errors is n-m. The “t Stat” is the
“observed” value of the t-statistic for each parameter, according to the null hypothesis that the true value
of the parameter is zero. The standard errors of the parameters can also be used with the t-distribution to
compute confidence intervals for the parameters. These are included in the Microsoft® Excel regression
output in the column “Lower 95%” and “Upper 95%” (not shown here).
EXAMPLE. For the regression example above, the observed value of the t-statistic for
the log KF parameter is
− 1.27 − 0
= 7 .9
0.161
The probability associated with the alternate hypothesis that the true value of log KF is
different from zero is determined from a two-sided t-test:
Pr{− 7.9 ≤ t 8 ≤ 7.9} = 0.99995
The difference between this value and unity, i.e. α, can be found in the Microsoft Excel
regression output under the “P-value” column. Clearly, the estimated value of log KF is
statistically very different from zero.
BIBLIOGRAPHY
Berthouex, P. M.; L. C. Brown. Statistics for Environmental Engineers. Lewis Publishers, 1994.
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Bevington, P. R.; D. K. Robinson. Data Reduction and Error Analysis for the Physical Sciences.
McGraw-Hill, Inc. 1992.
Box, G. E. P.; W. G. Hunter; J. S. Hunter. Statistics for Experimenters: An Introduction to Design,
Data Analysis, and Model Building. John Wiley & Sons, 1978.
Devore, J.; N. Farnum. Applied Statistics for Engineers and Scientists. Duxbury Press, 1999.
McBean, E. A.; F. A. Rovers. Statistical Procedures for Analysis of Environmental Monitoring Data
& Risk Assessment. Prentice Hall, 1998.
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